Title: Potential Theory on Stratified Lie Groups
by Mukund Madhav Mishra (Hansraj College) as part of Topological Groups
Lecture held in Elysium.
Stratified Lie groups form a special subclass of the class of nilpotent Lie groups. The Lie algebra of a stratified Lie group possesses a specific stratification (and hence the name), and an interesting class of anisotropic dilations. Among the linear differential operators of degree two, there exists a family that is well behaved with the automorphisms of the stratified Lie group, especially with the anisotropic dilations. We shall see that one such family of operators mimics the classical Laplacian in many aspects, except for the regularity. More specifically, these Laplace-like operators are sub-elliptic, and hence referred to as the sub-Laplacians. We will review certain interesting properties of functions harmonic with respect to the sub-Laplacian on a stratified Lie group, and have a closer look at a particular class of stratified Lie groups known as the class of Heisenberg type groups.